LOS ANGELES—Casey Kasem, a radio star whose smooth voice encouraged millions of American listeners to "keep your feet on the ground and keep reaching for the stars," died Sunday in a hospital outside Tacoma, Wash. He was 82.

The "American Top 40" host and voice of Shaggy on "Scooby-Doo, Where Are You!" had suffered from dementia, according to court papers. His death came after a protracted, often...